Math 2.0

Global analysis, analysis on manifolds Infinite-dimensional manifolds Example

Diffeomorphism group

Builds on DiffeomorphismSmooth manifoldVector fieldFréchet manifoldLie groupExponential map of a Lie group

Example

Let MM be a smooth manifold. The diffeomorphism group Diff⁡(M)\operatorname{Diff}(M) is the set of all diffeomorphisms f ⁣:M→Mf \colon M \to M, with composition as the operation. It is a group. The composition of two diffeomorphisms is a diffeomorphism, and composition is associative. The identity map id⁡M\id_M is the identity element, and the inverse of ff is the inverse map f−1f^{-1}, a diffeomorphism too.

When MM is compact, Diff⁡(M)\operatorname{Diff}(M) is an infinite-dimensional Lie group. It is a Fréchet manifold modelled on the space of smooth vector fields on MM, with smooth composition and inversion. Its Lie algebra is the space of vector fields; with the usual conventions, the bracket is the negative of the Lie bracket of vector fields.

Remarks

Unlike a finite-dimensional Lie group, Diff⁡(M)\operatorname{Diff}(M) has an exponential map, given by the time-one flow of a vector field, that need not be onto any neighbourhood of the identity. This happens already for M=S1M = S^1. So some results of finite-dimensional Lie theory fail for diffeomorphism groups.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

Show the path as a graph
Euclidean space, object, no evidence levelEuclidean spaceobjectExponential map of a Lie group, concept, no evidence levelExponential map of a …conceptFréchet manifold, concept, no evidence levelFréchet manifoldconceptFunction, concept, no evidence levelFunctionconceptLie group, concept, no evidence levelLie groupconceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptVector field, concept, no evidence levelVector fieldconceptDiffeomorphism, concept, no evidence levelDiffeomorphismconceptDiffeomorphism group, example, no evidence levelDiffeomorphism groupexample
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euclidean space object
  2. Exponential map of a Lie group concept
  3. Fréchet manifold concept
  4. Function concept
  5. Lie group concept
  6. Smooth manifold concept
  7. Total derivative concept
  8. Function of several variables concept
  9. Vector-valued function concept
  10. Differentiable function of several variables concept
  11. Vector field concept
  12. Diffeomorphism concept
  13. Diffeomorphism group example

Further reading

  1. Web https://en.wikipedia.org/wiki/Diffeomorphism
About this page
TagWTCNKST134
URIm2://WTCNKST134
Kindexample
WritingW0 stub
MSC58D05, 22E65, 57S05
Instance ofGroup
UsesDiffeomorphism, Smooth manifold, Vector field, Fréchet manifold, Lie group, Exponential map of a Lie group
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.